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Chapter 7 Liapunov's type equations
Chapter 7
Liapunov’s Type Equations 7.1
Solutions of Liapunov’s type equations
In this chapter H is a Hilbert space with a scalar product (., .), and A is a stable linear operator in H. Recall that a linear operator A is called stable if rs (A) < 1. Theorem 7.1.1 If A is a stable operator, then for any bounded linear operator C, there exists a linear operator WC , such that WC − A∗ WC A = C.
(1.1)
Moreover, WC =
∞ X
(A∗ )k CAk .
(1.2)
k=0
Thus, if C is strongly positive definite, then WC is strongly positive definite. Proof:
Since rs (A) < 1, the series (1.2) converges. Thus WC − A∗ WC A =
∞ X
(A∗ )k CAk − A∗
k=0
∞ X
(A∗ )k CAk A = C,
k=0
as claimed. Q. E. D.
Lemma 7.1.2 Let A be a stable operator and WC be a solution of (1.1), with a bounded linear operator C. Then 1 WC = 2π
Z
2π
(Ie−iω − A∗ )−1 C(Ieiω − A)−1 dω.
0
105
106 Proof:
CHAPTER 7. LIAPUNOV’S TYPE EQUATIONS Clearly, (Ie−iω − A∗ )−1 C(Ieiω − A)−1 = e−iω eiω (I − eiω A∗ )−1 C(I − e−iω A)−1 = ∞ X ∞ X
(eiω A∗ )k Ce−ijω Aj .
k=0 j=0
Integrate this equality term by term: Z
2π
(Ie−iω − A∗ )−1 C(Ieiω − A)−1 dω =
0 ∞ X ∞ Z X k=0 j=0
2π
e(k−j)iω (A∗ )k CAj dω = 2π
0
∞ X
(A∗ )k CAk = WC 2π.
k=0
As claimed. Q. E. D. In the sequel it is assumed that operator C is selfadjoint strongly positive definite. Thanks to (1.1) we have (WC Ah, Ah) = (A∗ WC Ah, h) = (WC h, h) − (Ch, h) (h ∈ H). That is, (WC Ah, Ah) < (WC h, h) (h ∈ H).
(1.3)
Moreover, from (1.1) it follows that C < WC
(1.4)
in the sense (Ch, h) < (WC h, h) (h ∈ H). Hence we get (h, h) ≤ (Ch, h) < (WC h, h). kC −1 k Consequently, (h, h) < kWC k(h, h) (h ∈ H), kC −1 k and thus, kC −1 kkWC k > 1.
(1.5)
Lemma 7.1.3 If equation (1.1) with C = C ∗ > 0 has a solution WC > 0, then the spectrum of A is located inside the unit disk.
7.2. BOUNDS FOR SOLUTIONS OF LIAPUNOV’S TYPE EQUATIONS 107 Proof: First let λ be an eigenvalue of A and u be the corresponding eigenvector. That is, Au = λu; hence, −(Cu, u) = ((A∗ WC A − WC )u, u) = (WC Au, Au) − (WC u, u) = |λ|2 (WC u, u) − (WC u, u) < 0 and since (WC u, u) > 0, it follows that |λ| < 1. Now let λ be a point of the continuous spectrum, such that |λ| = rs (A). Then for any > 0, there is a vector u, such that kuk = 1 and the vector v = Au − λu has the norm kvk ≤ (see (Ahiezer and Glazman, 1966, Section 93)). Thus (WC Au, Au) = |λ|2 (WC u, u) + (WC v, v) + 2Re λ(WC v, u). But |(WC v, v) + 2Re λ(WC v, u)| < 1 = kWC k(2 + 2|λ|). Thus 0 > −(Cu, u) = ((A∗ WC A − WC )u, u) = (WC Au, Au) − (WC u, u) ≤ |λ|2 (WC u, u) + 1 − (WC u, u). Now we can easily end the proof. Q. E. D.
7.2
Bounds for solutions of Liapunov’s type equations
In many applications, it is important to know bounds for the norm of the solution WC of the Liapunov type equation. Due to Lemma 7.1.2 Z kCk 2π kWC k ≤ k(eit I − A)−1 k2 dt. (2.1) 2π 0 Hence, kWC k ≤ kCk sup k(zI − A)−1 k2 . |z|=1
Lemma 7.2.1 Let operator A be stable. Then a solution W of the equation W − A∗ W A = I satisfies the inequality (W h, h) ≥
khk2 (h ∈ H). (kAk + 1)2
(2.2)
108 Proof:
CHAPTER 7. LIAPUNOV’S TYPE EQUATIONS By Lemma 7.1.2 we have Z 2π 1 (W h, h) = k(A − Ieiy )−1 hk2 dy (h ∈ H). 2π 0
But
khk khk ≥ . kA − Izk kAk + |z|
k(A − zI)−1 hk ≥ Thus,
1 (W h, h) ≥ 2π
Z 0
2π
khk2 dy , (kAk + 1)2
as claimed. Q. E. D.
7.3
Equivalent norms in a Hilbert space
Again consider the equation x(k + 1) = Ax(k) (k = 0, 1, ...)
(3.1)
with a bounded operator in H. Lemma 7.3.1 Let A be a stable operator. In addition, let C be Hermitian strongly positive definite. Then a solution x(k) of equation (3.1) satisfies the estimate. (WC x(k), x(k)) ≤ (1 −
1 )k (WC x(0), x(0)) kC −1 kkWC k
(k = 1, 2, ...) where WC is a solution of equation (1.1). Proof:
Put WC = W . Note that according to (1.5) kC −1 kkW k > 1. We have (W x(k), x(k)) = (W Ax(k − 1), Ax(k − 1)) = (A∗ W Ax(k − 1), x(k − 1)) = (W x(k − 1), x(k − 1)) − (Cx(k − 1), x(k − 1)).
But (Cx(k − 1), x(k − 1)) ≥ =
1 (x(k − 1), x(k − 1)) = kC −1 k
1 (W −1 W x(k − 1), x(k − 1)) ≥ kC −1 k 1 (W x(k − 1), x(k − 1)). kC −1 kkW k
7.3. EQUIVALENT NORMS IN A HILBERT SPACE
109
Hence the required result follows. Q. E. D. Define the scalar product (., .)W and the norm k.kW by (h, v)W = (WC h, v) and khkW = (WC h, h)1/2 . Recall that norms k.k and k.k1 are equivalent if there are positive constants c1 , c2 , such that c1 khk1 ≤ khk ≤ c2 khk1 for any h ∈ H. Clearly, norm k.kW is equivalent to norm k.k = k.kH . Thanks to the previous lemma, a solution x(.) of equation (3.1) satisfies the estimate kx(k)k2W ≤ (1 −
1 kC −1 kH kWC kH
)k kx(0)k2W (k = 1, 2, ...).
Denote ηA := sup k(zI − A)−1 k.
(3.2)
|z|=1
Then inequality (2.2) implies kx(k)k2W ≤ (1 −
kC −1 k
1 k 2 2 ) kx(0)kW (k = 1, 2, ...). H kCkH ηA
(3.3)
Let us return to the norm k.k = k.kH . Take C = I. Then Lemma 7.2.1 and (2.2) imply 1 1 kx(k)kH ≤ (1 − 2 )k/2 ηA kx(0)kH (k = 1, 2, ...). kAkH + 1 ηA We thus get Lemma 7.3.2 Let operator A be stable. Then k kAk kH ≤ MA νA (k ≥ 0)
where MA := (kAkH + 1)ηA and s νA :=
1−
1 2 . ηA
(3.4)
110
7.4
CHAPTER 7. LIAPUNOV’S TYPE EQUATIONS
Particular cases
Let A be a stable operator. Corollary 7.4.1 Let A be a Hilbert-Schmidt operator. Then inequality (3.4) holds with ∞ X g k (A) √ ηA ≤ k!(1 − rs (A))k+1 k=0 and ηA ≤
1 1 g 2 (A) exp [ + ]. 1 − rs (A) 2 2(1 − rs (A))2
These results follow from (3.2) and Theorem 4.2.1. Now let A ∈ C2p (H) (p = 2, 3, ...).
(4.1)
That is A, is a Neumann-Schatten operator. Then due to Theorem 4.4.1, sup kRλ (A)k ≤ ηp (A) |z|=1
where ηp (A) :=
p−1 X
(2N2p (A))m 1 (2N2p (A))2p exp [ + ]. m+1 (1 − rs (A)) 2 2(1 − rs (A))2p m=0
We thus get Corollary 7.4.2 Under condition (4.1), inequalities (3.4) hold with ηA ≤ ηp (A). Similarly quasi-Hermitian and quasiunitary operators can be considered with the help of the estimates for the resolvent presented in Chapter 4, in particular, (3.2) and Theorem 4.5.1 imply Corollary 7.4.3 Let A − A∗ be a Hilbert-Schmidt operator. Then inequality (3.4) holds with 1 1 gI2 (A) ηA ≤ exp [ + ]. 1 − rs (A) 2 2(1 − rs (A))2 Now let A be an n × n matrix. Then according to (3.2) and Corollary 3.2.2 ηA ≤ η1,n (A) :=
n−1 X
√
k=0
g k (A) . k!(1 − rs (A))k+1
Moreover, Theorems 3.2.3 and 3.2.4 imply ηA ≤ η2,n (A) :=
1 1 g 2 (A) [1 + (1 + )](n−1)/2 1 − rs (A) n−1 (1 − rs (A))2
7.4. PARTICULAR CASES and ηA ≤ η3,n (A) :=
111
√ (N2 (A) + n)n−1 Q (n > 1). n (n − 1)(n−1)/2 k=1 (1 − |λk (A)|)
Now from Lemma 7.3.2 it follows. Corollary 7.4.4 Let A be a stable n×n-matrix. Then inequalities (3.4) hold with ηA ≤ ηj,n (A) (j = 1, 2, 3).
Liapunov’s Type Equations 7.1
Solutions of Liapunov’s type equations
In this chapter H is a Hilbert space with a scalar product (., .), and A is a stable linear operator in H. Recall that a linear operator A is called stable if rs (A) < 1. Theorem 7.1.1 If A is a stable operator, then for any bounded linear operator C, there exists a linear operator WC , such that WC − A∗ WC A = C.
(1.1)
Moreover, WC =
∞ X
(A∗ )k CAk .
(1.2)
k=0
Thus, if C is strongly positive definite, then WC is strongly positive definite. Proof:
Since rs (A) < 1, the series (1.2) converges. Thus WC − A∗ WC A =
∞ X
(A∗ )k CAk − A∗
k=0
∞ X
(A∗ )k CAk A = C,
k=0
as claimed. Q. E. D.
Lemma 7.1.2 Let A be a stable operator and WC be a solution of (1.1), with a bounded linear operator C. Then 1 WC = 2π
Z
2π
(Ie−iω − A∗ )−1 C(Ieiω − A)−1 dω.
0
105
106 Proof:
CHAPTER 7. LIAPUNOV’S TYPE EQUATIONS Clearly, (Ie−iω − A∗ )−1 C(Ieiω − A)−1 = e−iω eiω (I − eiω A∗ )−1 C(I − e−iω A)−1 = ∞ X ∞ X
(eiω A∗ )k Ce−ijω Aj .
k=0 j=0
Integrate this equality term by term: Z
2π
(Ie−iω − A∗ )−1 C(Ieiω − A)−1 dω =
0 ∞ X ∞ Z X k=0 j=0
2π
e(k−j)iω (A∗ )k CAj dω = 2π
0
∞ X
(A∗ )k CAk = WC 2π.
k=0
As claimed. Q. E. D. In the sequel it is assumed that operator C is selfadjoint strongly positive definite. Thanks to (1.1) we have (WC Ah, Ah) = (A∗ WC Ah, h) = (WC h, h) − (Ch, h) (h ∈ H). That is, (WC Ah, Ah) < (WC h, h) (h ∈ H).
(1.3)
Moreover, from (1.1) it follows that C < WC
(1.4)
in the sense (Ch, h) < (WC h, h) (h ∈ H). Hence we get (h, h) ≤ (Ch, h) < (WC h, h). kC −1 k Consequently, (h, h) < kWC k(h, h) (h ∈ H), kC −1 k and thus, kC −1 kkWC k > 1.
(1.5)
Lemma 7.1.3 If equation (1.1) with C = C ∗ > 0 has a solution WC > 0, then the spectrum of A is located inside the unit disk.
7.2. BOUNDS FOR SOLUTIONS OF LIAPUNOV’S TYPE EQUATIONS 107 Proof: First let λ be an eigenvalue of A and u be the corresponding eigenvector. That is, Au = λu; hence, −(Cu, u) = ((A∗ WC A − WC )u, u) = (WC Au, Au) − (WC u, u) = |λ|2 (WC u, u) − (WC u, u) < 0 and since (WC u, u) > 0, it follows that |λ| < 1. Now let λ be a point of the continuous spectrum, such that |λ| = rs (A). Then for any > 0, there is a vector u, such that kuk = 1 and the vector v = Au − λu has the norm kvk ≤ (see (Ahiezer and Glazman, 1966, Section 93)). Thus (WC Au, Au) = |λ|2 (WC u, u) + (WC v, v) + 2Re λ(WC v, u). But |(WC v, v) + 2Re λ(WC v, u)| < 1 = kWC k(2 + 2|λ|). Thus 0 > −(Cu, u) = ((A∗ WC A − WC )u, u) = (WC Au, Au) − (WC u, u) ≤ |λ|2 (WC u, u) + 1 − (WC u, u). Now we can easily end the proof. Q. E. D.
7.2
Bounds for solutions of Liapunov’s type equations
In many applications, it is important to know bounds for the norm of the solution WC of the Liapunov type equation. Due to Lemma 7.1.2 Z kCk 2π kWC k ≤ k(eit I − A)−1 k2 dt. (2.1) 2π 0 Hence, kWC k ≤ kCk sup k(zI − A)−1 k2 . |z|=1
Lemma 7.2.1 Let operator A be stable. Then a solution W of the equation W − A∗ W A = I satisfies the inequality (W h, h) ≥
khk2 (h ∈ H). (kAk + 1)2
(2.2)
108 Proof:
CHAPTER 7. LIAPUNOV’S TYPE EQUATIONS By Lemma 7.1.2 we have Z 2π 1 (W h, h) = k(A − Ieiy )−1 hk2 dy (h ∈ H). 2π 0
But
khk khk ≥ . kA − Izk kAk + |z|
k(A − zI)−1 hk ≥ Thus,
1 (W h, h) ≥ 2π
Z 0
2π
khk2 dy , (kAk + 1)2
as claimed. Q. E. D.
7.3
Equivalent norms in a Hilbert space
Again consider the equation x(k + 1) = Ax(k) (k = 0, 1, ...)
(3.1)
with a bounded operator in H. Lemma 7.3.1 Let A be a stable operator. In addition, let C be Hermitian strongly positive definite. Then a solution x(k) of equation (3.1) satisfies the estimate. (WC x(k), x(k)) ≤ (1 −
1 )k (WC x(0), x(0)) kC −1 kkWC k
(k = 1, 2, ...) where WC is a solution of equation (1.1). Proof:
Put WC = W . Note that according to (1.5) kC −1 kkW k > 1. We have (W x(k), x(k)) = (W Ax(k − 1), Ax(k − 1)) = (A∗ W Ax(k − 1), x(k − 1)) = (W x(k − 1), x(k − 1)) − (Cx(k − 1), x(k − 1)).
But (Cx(k − 1), x(k − 1)) ≥ =
1 (x(k − 1), x(k − 1)) = kC −1 k
1 (W −1 W x(k − 1), x(k − 1)) ≥ kC −1 k 1 (W x(k − 1), x(k − 1)). kC −1 kkW k
7.3. EQUIVALENT NORMS IN A HILBERT SPACE
109
Hence the required result follows. Q. E. D. Define the scalar product (., .)W and the norm k.kW by (h, v)W = (WC h, v) and khkW = (WC h, h)1/2 . Recall that norms k.k and k.k1 are equivalent if there are positive constants c1 , c2 , such that c1 khk1 ≤ khk ≤ c2 khk1 for any h ∈ H. Clearly, norm k.kW is equivalent to norm k.k = k.kH . Thanks to the previous lemma, a solution x(.) of equation (3.1) satisfies the estimate kx(k)k2W ≤ (1 −
1 kC −1 kH kWC kH
)k kx(0)k2W (k = 1, 2, ...).
Denote ηA := sup k(zI − A)−1 k.
(3.2)
|z|=1
Then inequality (2.2) implies kx(k)k2W ≤ (1 −
kC −1 k
1 k 2 2 ) kx(0)kW (k = 1, 2, ...). H kCkH ηA
(3.3)
Let us return to the norm k.k = k.kH . Take C = I. Then Lemma 7.2.1 and (2.2) imply 1 1 kx(k)kH ≤ (1 − 2 )k/2 ηA kx(0)kH (k = 1, 2, ...). kAkH + 1 ηA We thus get Lemma 7.3.2 Let operator A be stable. Then k kAk kH ≤ MA νA (k ≥ 0)
where MA := (kAkH + 1)ηA and s νA :=
1−
1 2 . ηA
(3.4)
110
7.4
CHAPTER 7. LIAPUNOV’S TYPE EQUATIONS
Particular cases
Let A be a stable operator. Corollary 7.4.1 Let A be a Hilbert-Schmidt operator. Then inequality (3.4) holds with ∞ X g k (A) √ ηA ≤ k!(1 − rs (A))k+1 k=0 and ηA ≤
1 1 g 2 (A) exp [ + ]. 1 − rs (A) 2 2(1 − rs (A))2
These results follow from (3.2) and Theorem 4.2.1. Now let A ∈ C2p (H) (p = 2, 3, ...).
(4.1)
That is A, is a Neumann-Schatten operator. Then due to Theorem 4.4.1, sup kRλ (A)k ≤ ηp (A) |z|=1
where ηp (A) :=
p−1 X
(2N2p (A))m 1 (2N2p (A))2p exp [ + ]. m+1 (1 − rs (A)) 2 2(1 − rs (A))2p m=0
We thus get Corollary 7.4.2 Under condition (4.1), inequalities (3.4) hold with ηA ≤ ηp (A). Similarly quasi-Hermitian and quasiunitary operators can be considered with the help of the estimates for the resolvent presented in Chapter 4, in particular, (3.2) and Theorem 4.5.1 imply Corollary 7.4.3 Let A − A∗ be a Hilbert-Schmidt operator. Then inequality (3.4) holds with 1 1 gI2 (A) ηA ≤ exp [ + ]. 1 − rs (A) 2 2(1 − rs (A))2 Now let A be an n × n matrix. Then according to (3.2) and Corollary 3.2.2 ηA ≤ η1,n (A) :=
n−1 X
√
k=0
g k (A) . k!(1 − rs (A))k+1
Moreover, Theorems 3.2.3 and 3.2.4 imply ηA ≤ η2,n (A) :=
1 1 g 2 (A) [1 + (1 + )](n−1)/2 1 − rs (A) n−1 (1 − rs (A))2
7.4. PARTICULAR CASES and ηA ≤ η3,n (A) :=
111
√ (N2 (A) + n)n−1 Q (n > 1). n (n − 1)(n−1)/2 k=1 (1 − |λk (A)|)
Now from Lemma 7.3.2 it follows. Corollary 7.4.4 Let A be a stable n×n-matrix. Then inequalities (3.4) hold with ηA ≤ ηj,n (A) (j = 1, 2, 3).